On the Local Separation of E and B Polarization Patterns in CMBR

This work examines the small-scale E and B patterns of the Cosmic Microwave Background Radiation (CMBR) polarization anisotropy. Particularly, we address the topological natures of the E and B modes, and how one may make use of the local measurements of Stokes parameters to separate these two modes in the real space. The analysis of a local map in the Fourier space for separating the E and B modes can be an ill-posed problem, due to the non-periodic boundary condition in the map for each mode. A strategy for the E − B separation in a local map is through an appropriate projection of the polarization tensor field into a unique vector field, which naturally contains a curl-free E component and a divergence-free B component. An integral method, equivalent to the real-space top-hat filtering, is proposed for extracting the averaged ”charge” and ”current” of the E and B modes separately. The real-space top-hat filter function exhibits, in its power spectrum, an oscillation of a well-defined frequency comparable to the filter size. It is pointed out that when the inverse filter size is chosen properly, matching the oscillation period in the E-mode power spectrum, the sensitivity of the E-mode detection can be made to be significantly improved. Subject headings: Cosmic Microwave Background — Polarization